Proposition 6.1

Suppose that , and that for all conjugates , for some . Then is a beta number (and hence ).

Proof

The points of the orbit correspond to the points in the lattice . By the estimates on the conjugates just given, these n points lie in a cube of volume , and since the points of the lattice have a constant density in , there are points of the lattice in . By the box principle, for sufficiently large n we must have for some and hence the orbit is finite. Once we know the orbit is finite, the conjugates lie in a finite set, and so certainly .